To this end, it is necessary that some part of it should be conceived to be cut off by a plane parallel to the fracture, so that the momentum of the part retrenched be to its resistance in the seine ratio as the momentum of the whole is to its resistance. These four powers act by arms of levers peculiar to themselves, and are proportional in the whole, and in each part, of a solid'of equal resistance. From this prdportion, Varignon deduces two solids, which shall resist equally in all their parts, or be no more liable to break iu one part than in another. Galileo had previously found one of these, which is that in which the sides are parabolical : the other, found by Varignon, is in the form of a trumpet, which is to be fixed into the wall by its greater end ; so that its magnitude, or weight, is always diminished in proportion as its length, or the arm of the lever by which it acts, is increased. It is remarkable, that, however different the two systems may be, the solids of equal resistance are the same in both.
The following is a general synopsis of the most important results which have been drawn by difThrent writers on this subject, both practical and theoretical.
1. The resistance of a beam or bar, to a fracture, by a force acting laterally, is as the solid made by a section of the beam in the place where the force is applied, into the distance of its centre of gravity from the point or line where the breach will end.
2. In square beams, the lateral strengths are as the cubes of their breadths or depths.
3. In cylindric beams the resistances or strengths are as the cubes of the diameters.
4. In rectangular beams, the lateral strengths are con jointly as the breadths and squares of the depths.
5. The lateral resistance of any beams, whose sections are similar figures and alike placed, are as the cubes of the like dimensions of those figures.• 6. The lateral strength of a beam, with its narrower face upwards, is to its strength with the broader •ice upwards, as the breadth of the broader face to the breadth of the narrower.
7. The lateral strengths of prismatic beams of the same materials, are as the areas of the sections, and the distance of their centre of gravity, directly, and as their lengths and weights reciprocally.
S. When the beam is fixed at both ends, the same property has place, except that, in this case, we must consider the beam as only half the length of the former.
9. Cylinders and square prisms have their lateral strengths proportional to the cubes of their diameters, or depths, directly, and their lengths and weights inversely.
10. Similar prisms and cylinders have their strength inversely proportional to their linear dimensions.
The following results are wholly drawn from experi ments on different substances, by Emerson and other writers, by means of which the propositions stated in the preceding part of this article may be submitted to computations.
A cylindric rod of good clean fir, of an inch circumference, drawn in length, will bear at its extremity 4001b. ; and a spear of fir, of two inches diameter, will bear about seven ton weight. A rod of good iron, of an inch circumference, will bear nearly three ton weight. A good hempen rope of an inch circumference, will bear 1,000lb. at its extremity. Hence Emerson concludes, that if it rod of fir, or a rope, or a rod of iron, of d inches diameter, were to lift a quarter of the extreme weight that they would support, then The fir would bear . . d' hundred weight. The rope 22 ditto.
The iron y tolls.
To these results we may add, from the experiments and investigations of Professor Robison, that a prism of white marble, an inch square and a foot about 500 lb. And that, from the various authors he has collected, the cohesive force of a square inch of gold, when cast, is about 20,600 lb ; of silver, 40,000 lb. ; cast-iron from 40,000 to 60,000 lb. ; wrought iron from 60,000 to 90,000; soft steel, 12,000 lb. ; razor steel, 15,000 lb. ; oak and beech, in the direction of their fibres, from to 17,000 lb. ; willow, 1•,000 ; cedar, 5,000 lb. ; fir, 8,000 lb. ; ivory, 16,000 lb.; bone, 5,000 lb; rope, 20,000 lb. And a cylinder, an inch in diameter, loaded to one-fourth, will carry, if of iron, 135 cwt. ; of rope, 22 cwt. ; oak, 14 cwt. ; and fir. 9 cwt.
The resistance of some metals is doubled, or tripled, by the operation of forging and wire-drawing ; and the cohesive, as well as the repulsive force of wood, is often increased by moderate compression. Oak will suspend much more than fir ; but lir will support twice as much as oak ; which differ ence is supposed to arise from the curvature of the fibres of oak ; yet oak has been known to support, Stone safety, more than two tons for every square inch. Stone will support from 250 to 850,000 pounds, on a foot square ; brick, 300 lb.; and sometimes they are practically made to support one-sixth as much. Stone is said to be capable of bearing a much greater weight in that position in which it is found in the quarry, than in any other position.